We have investigated the atomic structure of the twofold surface of the decagonal Al-Cu-Co quasicrystal using scanning tunneling microscopy and low-energy electron diffraction. We have found that most of the surface features can be interpreted using the bulk-structure model proposed by Deloudi and Steurer (S. Deloudi, Ph.D. thesis, ETH, Zürich, 2008). The surface consists of terraces separated by steps of various heights. Step heights and steps sequences match with the thickness and the stacking sequence of blocks of layers separated by gaps in the model. These blocks of layers define possible surface terminations consisting of periodic atomic rows which are aperiodically stacked. These surface terminations are dense (∼10 at./nm2) and are of three types. The first two types are pure or almost pure Al while the third one contains 30–40 at. % of transition-metal atoms. Experimentally, we observe three different types of fine structures on terraces, which can be interpreted using the three possible types of bulk terminations. Terraces containing transition metals exhibit a strong bias dependency and present a doubling of the basic 0.42 nm periodicity, in agreement with the 0.84 nm superstructure of the bulk. In addition, a high density of interlayer phason defects is observed on this surface that could contribute to the stabilization of this system through configurational entropy associated with phason disorder.